Question: Find the sum of $-50 + (-44) + (-38) +... + 2038 + 2044$.
Explanation: Getting started We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $6$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-50})$ and the last term $(a_n = {2044})$ are given in the question. We need to find $n$ (the number of terms). Step 1: Find $n$ (the number of terms) The sequence increases by $2044 - (-50) = 2094$ from the first term to the last term. Because the sequence increases by $6$ each time, it takes $\dfrac{2094}{6} = 349$ term to get from the first term to the last term. We still need to count the first term, so there are $349 + 1 = {350}$ terms in the sequence. In other words, $n = {350}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{350}}&= \dfrac {\left({-50} + {2044} \right)}{2} \cdot {350} \\\\ S_{{350}} &= 997 \left(350\right) \\\\ S_{{350}} &= 348{,}950\end{aligned}$ The answer $ 348{,}950 $